Soliton Research Articles

Numerical simulation of coherent mid-infrared supercontinuum generation based on carbon chloride core tapered photonic crystal fibers

We numerically studied coherent mid-infrared (MIR) supercontinuum (SC) generated by filling carbon chloride (CCl4, C2Cl4) into the tapered photonic crystal fiber (PCF). The mechanism involves nonlinear soliton compression, Raman soliton self-frequency shift and dispersive wave generated in a carbon chloride core tapered PCF pumped by a 1.98 μm femtosecond laser. Numerical solution of the generalized nonlinear Schrodinger equation shows that 1.15–5.5 μm (CCl4-core tapered PCF) and 1.15–5.7 μm (C2Cl4-core tapered PCF) coherent MIR-SC sources can be realized. Our results show that the above liquid-core tapered PCFs are promising nonlinear media for constructing broadband coherent MIR-SC light sources.

Quasi-Grammian solutions of the coupled Gerdjikov–Ivanov equation

We construct the N-fold standard binary Darboux transformation (bDT) for the coupled Gerdjikov–Ivanov (cGI) equation. Then, using the bDT presented in this article, we explicitly construct the exact solutions of the cGI equation in terms of quasi-Grammians. As applications of the bDT, we present various particular solutions for the cGI equation, including soliton, breather, and rogue wave solutions.

Multiple Soliton Solutions for Coupled Modified Korteweg–de Vries (mkdV) with a Time-Dependent Variable Coefficient

In this manuscript, we implement analytical multiple soliton wave and singular soliton wave solutions for coupled mKdV with a time-dependent variable coefficient. Based on the similarity transformation and Hirota bilinear technique, we construct both multiple wave kink and wave singular kink solutions for coupled mKdV with a time-dependent variable coefficient. We implement the Hirota bilinear technique to compute analytical solutions for the coupled mKdV system. Such calculations are made by using a software with symbolic computation software, for instance, Maple. Recently some researchers used Maple in order to show that the bilinear method of Hirota is a straightforward technique which can be used in the approach of differential, nonlinear models. We analyzed whether the experiments proved that the procedure is effective and can be successfully used for many other mathematical models used in physics and engineering. The results of this study display that the profiles of multiple-kink and singular-kink soliton types can be efficiently controlled by selecting the particular form of a similar time variable. The changes in the solitons based on the changes in the arbitrary function of time allows for more applications of them in applied sciences.

Traveling wave solutions to the nonlinear space–time fractional extended KdV equation via efficient analytical approaches

In this paper, the nonlinear space-time fractional extended Korteweg-de Vries (F-eKdV) equation involving the conformable fractional derivative (CFD) is considered. Bernoulli and Riccati equations as a simplest equation method together with applying modified traveling wave transformation are used to obtain the exact solution of F-eKdV equation. Also, by exp⁡(−Φ(ζ))-expansion method, different kind of exact traveling wave solutions are gained, including periodic-singular, dark-singular and plane-wave solutions. As a result, several soliton solutions of the proposed equation are revealed. These novel soliton waves are constructed using the software package Maple with some given parametric values and displayed graphically in 2-dimensional (2D) and 3-dimensional (3D) plots by using MATLAB. To illustrate how the fractional operator affects the results, some of the acquired solutions are presented for various values of the fractional order α and also compared to their exact solutions in classical case (i.e. α=1). In the given graphical representations, the physical meaning of the reported solutions is illustrated. From the obtained results, it is observed that the methods are effective and dependable to solve nonlinear fractional partial differential equations (NFPDEs).

Spatial self-bending soliton phenomenon of (2+1) dimensional bidirectional Sawada-Kotera equation

In this study, we delve into the phenomenon of spatial self-bending solitons within the context of the (2+1)-dimensional bidirectional Sawada-Kotera equation. By employing the Hirota bilinear method, the bilinear form of the (2+1)-dimensional bidirectional Sawada-Kotera equation can be obtained. Through specific parameter constraints (eAjs=0), we establish the N-soliton solution and subsequently derive the spatial self-bending soliton. The curvature of this spatial self-bending soliton is also elucidated. Additionally, we explore the interaction between the spatial self-bending soliton and respiratory waves, along with the interaction between the spatial self-bending soliton and higher-order lump waves through the method of the long wave limit.

Unraveling the Dynamics of Singular Stochastic Solitons in Stochastic Fractional Kuramoto–Sivashinsky Equation

This work investigates the complex dynamics of the stochastic fractional Kuramoto–Sivashinsky equation (SFKSE) with conformable fractional derivatives. The research begins with the creation of singular stochastic soliton solutions utilizing the modified extended direct algebraic method (mEDAM). Comprehensive contour, 3D, and 2D visual representations clearly depict the categorization of these stochastic soliton solutions as kink waves or shock waves, offering a clear description of these soliton behaviors within the context of the SFKSE framework. The paper also illustrates the flexibility of the transformation-based approach mEDAM for investigating soliton occurrence not only in SFKSE but also in a wide range of nonlinear fractional partial differential equations (FPDEs). Furthermore, the analysis considers the effect of noise, specifically Brownian motion, on soliton solutions and wave dynamics, revealing the significant influence of randomness on the propagation, generation, and stability of soliton in complex stochastic systems and advancing our understanding of extreme behaviors in scientific and engineering domains.

Nonlinear dissipative wave trains in a system of self-propelled particles

The paper addresses the existence of modulated nonlinear periodic wave trains in a system of self-propelled particles (SPPs). The reductive perturbation method reduces the model hydrodynamics equations to a one-dimensional (1D) complex Ginzburg-Landau (CGL) equation. The modulational instability (MI) phenomenon is studied, where an expression for the instability growth rate is proposed. The latter is used to discuss regions of parameters where trains of solitonic waves are likely to be obtained. This is highly influenced by the values of the variances of Gaussian noise in self-diffusion and binary collision. Solutions for the CGL equations are also studied via the Porubov technique, using a combination of Jacobi and Weierstrass elliptic functions. Wave propagation in the self-propelled particles flock includes modulated nonlinear wave trains, nonlinear spatially localized periodic patterns, and continuous waves.

Precise invariant travelling wave soliton solutions of the Nizhnik–Novikov–Veselov equation with dynamic assessment

The (2 + 1)-dimensional Nizhnik–Novikov–Veselov equation is a significant model that describes the behaviour of conserved scalar nucleons and their connection to neutral scalar masons. This system exhibits self-formation and depends on various free parameters. It is commonly used to study the dynamics of scalar nucleons and neutral scalar masons. This research article used an extended direct algebraic technique to obtain exact travelling wave solutions for the Nizhnik–Novikov–Veselov system. The obtained soliton solutions include various types, such as multi-periodic, periodic, plane, singular, bright, dark, and flat kink-type wave solitons. We present these soliton solutions graphically by varying the involved parameters using the advanced software program Wolfram Mathematica. The graphical representations in 3D, contour, and 2D surfaces allow us to visualise the behaviour of the solutions as the parameters change and the physical interpretation for these solutions is obtained. Additionally, we conduct a sensitivity analysis to examine the wave profiles of the newly designed dynamical framework. The results of this analysis demonstrate the reliability and efficiency of the proposed method, which can be applied to find closed-form travelling wave solitary solutions for a wide range of nonlinear evolution equations.

On new diverse variety analytical optical soliton solutions to the perturbed nonlinear Schrödinger equation

This study focuses on investigating the optical soliton solutions for the perturbed nonlinear Schrödinger equation involving Kerr nonlinearity. It should be noted that the non-integrable nature of the nonlinear Schrödinger equation becomes apparent when the Kerr law nonlinearity is absent. This absence of integrability significantly obstructs finding exact solutions. Our research utilizes an innovative analytical technique, the modified generalized exponential rational function method, to derive new soliton solutions. The new soliton solutions we obtained are quite versatile and practical for real-world applications. They are expressed in terms of elementary functions such as exponentials, trigonometric functions, and hyperbolic functions. This simplicity allows for easy comprehension and straightforward utilization in various scenarios. We present a variety of visualizations and graphs to enhance the understanding of the physical behaviors exhibited by the new soliton solutions. The graphical outcomes give insights into soliton evolution and dynamics under various conditions. Our study shows the analytical technique’s efficacy in producing new soliton solutions for the perturbed nonlinear Schrödinger equation. We explore the possibility of expanding the method to discover solutions for alternative nonlinear partial differential equations that arise within soliton theory and nonlinear optics. The analytical approach offers a systematic framework for obtaining new soliton solutions for a broad class of nonlinear wave equations. Our employed methodologies hold also the potential to significantly advance the field and contribute to the development of new methodologies for tackling these challenging equations.

Some Latest Families of Exact Solutions to Date–Jimbo–Kashiwara–Miwa Equation and Its Stability Analysis

The present study demonstrates the derivation of new analytical solutions for the Date–Jimbo–Kashiwara–Miwa equation utilizing two distinct methodologies, specifically the modified Kudryashov technique and the (g′)-expansion procedure. These innovative concepts employ symbolic computations to provide a dynamic and robust mathematical procedure for addressing a range of nonlinear wave situations. Additionally, a comprehensive stability analysis is performed, and the acquired results are visually represented through graphical representations. A comparison between the discovered solutions and those already found in the literature has also been performed. It is anticipated that the solutions will contribute to the existing literature related to mathematical physics and soliton theory.

New exact soliton and periodic wave solutions of the nonlinear fractional evolution equations with additional term

This paper presents exact solutions for the fractional Korteweg–de Vries equation and the fractional modified Korteweg–de Vries equation with additional term using the functional variable method. The main advantage of the proposed functional variable method over other methods is that it provides more accurate traveling wave solutions with additional free parameters. As a result, soliton and periodic wave solutions are obtained. It is shown that the calculations in the functional variable method are very simple and easy to understand, and this method is very effective for handling nonlinear fractional equations. Exact solutions are great importance in revealing the internal mechanism of physical phenomena. This method presents a wider applicability for handling nonlinear fractional-order wave equations.

Exploring lump soliton solutions and wave interactions using new Inverse $$(G'/G)$$-expansion approach: applications to the (2+1)-dimensional nonlinear Heisenberg ferromagnetic spin chain equation

Exploring lump soliton solutions and wave interactions using new Inverse $$(G'/G)$$-expansion approach: applications to the (2+1)-dimensional nonlinear Heisenberg ferromagnetic spin chain equation

Експертна оцінка критеріїв вибору методів знешкодження зарядів виведених з експлуатації твердопаливних ракет

The subject of study in this article is the processes that occur during the disposal of solid fuel rockets, methods of disposal, and criteria for choosing an effective method of disposal. This study identified the most significant criteria that determine the choice of an effective method for the disposal of decommissioned solid-fuel rocket charges. The task: to consider the risks of the man-made impact of the process of disposal of solid-fuel intercontinental ballistic missiles on the environment, associated with the emission of harmful substances into the atmosphere, the scattering of fragments, the occurrence of shock air, and seismic and soliton waves; analyze the advantages and disadvantages of existing methods of destruction and use of solid propellant charges of rocket engines; expertly evaluate the criteria for the selection of disposal methods; and a priori ranking of criteria. The method of a priori ranking was chosen as the method of solving the given problem. The following results were obtained. The degree of consistency of the experts’ conclusions was assessed using the concordance coefficient. The hypothesis regarding the significance of the concordance coefficient was tested on the basis of a comparison of the calculated value of the concordance coefficient with the tabulated value at the given values of the significance level and the number of degrees of freedom. The most significant factors that should be taken into account when choosing a disposal method are identified (economic efficiency; the possibility of reusing the fuel that is removed; manufacturability; the probability of emergency situations during the implementation of the method; environmental safety; energy intensity), and factors that can be excluded from further analysis (the need for additional special systems, the degree of mastery of the method). Conclusions. The scientific novelty of the obtained results is as follows: the formalization and mathematical processing of a priori data on the subject of research was conducted, and a comparative assessment of the degree of influence of various criteria on known and promising methods of disposal of solid rocket fuel charges is given. According to the preliminary evaluation, the set of significant selection criteria is most fully met by the cryogenic washing method, the application of which allows the use of solid fuel and crushed substances extracted from solid fuel charges to improve the characteristics of modern detonation engines, to improve the characteristics of the technological processes of applying special refractory coatings, and to organize highly efficient combustion processes in the detonation mode of fuel with the production of thermal and electrical energy.

Data-driven localized waves and parameter discovery in the massive Thirring model via extended physics-informed neural networks with interface zones

In this paper, we study data-driven localized wave solutions and parameter discovery in the massive Thirring (MT) model via the deep learning in the framework of physics-informed neural networks (PINNs) algorithm. Abundant data-driven solutions including soliton of bright/dark type, breather and rogue wave are simulated accurately and analyzed contrastively with relative and absolute errors. For higher-order localized wave solutions, we employ the extended PINNs (XPINNs) with domain decomposition to capture the complete pictures of dynamic behaviors such as soliton collisions, breather oscillations and rogue-wave superposition. In particular, we modify the interface line in domain decomposition of XPINNs into a small interface zone and introduce the pseudo initial, residual and gradient conditions as interface conditions linked adjacently with individual neural networks. Then this modified approach is applied successfully to various solutions ranging from bright-bright soliton, dark-dark soliton, dark-antidark soliton, general breather, Kuznetsov-Ma breather and second-order rogue wave. Experimental results show that this improved version of XPINNs reduce the complexity of computation with faster convergence rate and keep the quality of learned solutions with smoother stitching performance as well. For the inverse problems, the unknown coefficient parameters of linear and nonlinear terms in the MT model are identified accurately with and without noise by using the classical PINNs algorithm.

Soliton interactions and Mach reflection in gas bubbles–liquid mixtures

In this study, we assume that blood is assumed to be a viscoelastic and incompressible homogeneous media in which several uniform sized oxygen bubbles are uniformly distributed. Based on this, we establish a (3 + 1)-dimensional modified Kadomtsev–Petviashvili (mKP) equation to describe the long nonlinear pressure waves in the gas bubbles–liquid mixtures. Using bell polynomials, a new bilinear form of the mKP equation is constructed, and then the one- and two-soliton solutions of the equation are obtained by the Hirota method. Via the one-soliton solutions, parametric conditions of the existence of shock wave, elevation and depression solitons, and the Mach reflection characters in the mixtures are discussed. Soliton interactions have been discussed on the basis of the two-soliton solutions. We find that the (i) parallel elastic interactions can exist between the shock and elevation solitons; (ii) oblique elastic interactions can exist between the (a) shock and depression solitons and (b) the elevation and depression solitons; and (iii) oblique inelastic interactions can exist between the two depression solitons.

Nonlinear gravity waves in the channel covered by broken ice

The two-dimensional nonlinear problem of a steady flow in a channel covered by broken ice with an arbitrary bottom topography including a semi-circular obstruction is considered. The mathematical model is based on the velocity potential theory accounting for nonlinear boundary conditions on the bottom of the channel and at the interface between the liquid and the layer of the broken ice, which are coupled through a numerical procedure. A mass loading model together with a viscous layer model is used to model the ice cover. The integral hodograph method is employed to derive the complex velocity potential of the flow, which contains the velocity magnitude at the interface in explicit form. The coupled problem is reduced to a system of integral equations in the unknown velocity magnitude at the interface, which is solved numerically using a collocation method. Case studies are conducted both for the subcritical and for the supercritical flow regimes in the channel. For subcritical flows, it is found that the ice cover allows for generating waves with amplitudes larger than those that may exist in the free surface case; the ice cover prevents the formation of a cusp and extends the solution to larger obstruction heights on the bottom. For supercritical flow regimes, the broken ice significantly affects the waveform of the soliton wave making it gentler. The viscosity factor of the model apparently governs the wave attenuation.

WITHDRAWN: The construction of first integral, soliton wave solutions, chaos analysis and sensitive demonstration of Konopelchenko–Dubrovsky equation

WITHDRAWN: The construction of first integral, soliton wave solutions, chaos analysis and sensitive demonstration of Konopelchenko–Dubrovsky equation

Passive mode locking erbium-doped fiber laser using V2O5 polyethylene glycol saturable absorber

An ultrashort pulse erbium-doped fiber laser (EDFL) in anomalous group delay dispersion (GDD) has been proven to produce a soliton wave production at 1596 nm. The mode-locking operation was generated by employing a vanadium pentoxide-polyethylene glycol (V<sub>2</sub>O<sub>5</sub>)-(PEG) film as absorber for all-fiber ring setup. Under anomalous dispersion, the soliton mode-locked laser produced a peak wavelength with 2.7 nm spectral bandwidth and Kelly sidebands. Under this condition, we obtained pulse energy of 210 nJ and pulse width of 1.40 ps. The maximum peak electrical power of 150 kW was calculated at the maximum pump power. These findings shows that the V<sub>2</sub>O<sub>5</sub>-PEG film can be a good saturable absorber (SA) device in generating stable mode-locking fiber laser at the 1.55- area.

Modulation instability, bifurcation analysis and solitonic waves in nonlinear optical media with odd-order dispersion

In this paper, modulation instability, bifurcation analysis, and soliton solutions are investigated in nonlinear media with odd-order dispersion terms. A generalized nonlinear Schrödinger equation with fourth-order dispersion and cubic-quintic nonlinearity is considered. A linear analysis method is used to derive an expression of the modulation instability spectrum, and the effects of the fourth-order and group velocity dispersion are pointed out on the modulation instability bands. The results show that for negative values of the fourth order in a normal dispersion regime, the modulation instability vanishes. Another important aspect is revealed when the fourth-order dispersion gets positive and has high values; additional bands emerge to enlarge the bandwidths of the modulation instability. To further confirm the effects of the odd-order dispersion in the nonlinear structure, the bifurcation, phase portraits, and chaotic behaviors have been investigated to show how instability arises in the nonlinear structure. Using numerical simulation, in particular the Runge-Kutta algorithm, the sensitivity of nonlinear systems is pointed out to confirm an unstable behavior. This investigation confirms once again the fact that the modulation spectrum is sensitive to higher-order dispersion in normal and anomalous dispersion regimes. A traveling wave hypothesis is employed to lead to the direct integration of the nonlinear system, and some specific soliton solutions are extracted. For particular constraint conditions on the discriminant, bright and dark solitons as well as Jacobi elliptic function solutions emerge. The obtained results could be used to improve the transmission signal via the optical fiber and secure the data in communication systems.

Higher-order rogue waves and dispersive solitons of a novel P-type (3+1)-D evolution equation in soliton theory and nonlinear waves

Higher-order rogue waves and dispersive solitons of a novel P-type (3+1)-D evolution equation in soliton theory and nonlinear waves